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The cut cone III: On the role of triangle facets

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Abstract

The cut polytopeP n is the convex hull of the incidence vectors of the cuts (i.e. complete bipartite subgraphs) of the complete graph onn nodes. A well known class of facets ofP n arises from the triangle inequalities:x ij +x ik +x jk ≤ 2 andx ij -x ik -x jk ≤ 0 for 1 ≤i,j, k ≤n. Hence, the metric polytope Mn, defined as the solution set of the triangle inequalities, is a relaxation ofP n . We consider several properties of geometric type for Pn, in particular, concerning its position withinM n . Strengthening the known fact ([3]) thatP n has diameter 1, we show that any set ofk cuts,k ≤ log2 n, satisfying some additional assumption, determines a simplicial face ofM n and thus, also, ofP n . In particular, the collection of low dimension faces ofP n is contained in that ofM n . Among a large subclass of the facets ofP n , the triangle facets are the closest ones to the barycentrum of Pn and we conjecture that this result holds in general. The lattice generated by all even cuts (corresponding to bipartitions of the nodes into sets of even cardinality) is characterized and some additional questions on the links between general facets ofP n and its triangle facets are mentioned.

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Authors and Affiliations

  1. CNRS, Université Paris VII, 17 Passage de l’Industrie, F-75010, Paris, France

    Michel Deza

  2. CNRS, LAMSADE, Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775, Paris Cedex 16, France

    Monique Laurent

  3. Department of Applied Mathematics, Charles University, Malostranske n.25, 11800, Praha 1, Czechoslovakia

    Svatopluk Poljak

Authors
  1. Michel Deza
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  2. Monique Laurent
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  3. Svatopluk Poljak
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Deza, M., Laurent, M. & Poljak, S. The cut cone III: On the role of triangle facets. Graphs and Combinatorics 9, 135–152 (1993). https://doi.org/10.1007/BF02988300

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  • DOI: https://doi.org/10.1007/BF02988300

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